HVAC system relay autotuning and verification

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent Application Ser. No. 61/620,584, filed on Apr. 5, 2012. The entirety of this priority document is incorporated herein by reference.

BACKGROUND

Proportional-integral (PI) feedback control can be used to regulate disturbances and achieve desired setpoints in HVAC systems. One example of a component in an HVAC system that can use such PI feedback control is a fan coil unit (FCU), which can regulate fan speed, cooling/heating medium flow, and/or other variables to meet desired temperature of a zone. The PI parameters, proportional gain and integral time can, however, be difficult to determine and pre-set, prior to commissioning of the HVAC system. Further, when implemented, such HVAC systems and components can be affected by a variety of time-dependent variables that often make a single set of PI parameters unsuitable for indefinite use. As a result, PI parameter re-tuning is often desired, allowing the controller to respond to changes in the characteristics of the system.

To provide such dynamic system control, the HVAC system controllers may be re-tuned while online. Tuning may be manual, proceeding by trial-and-error, or may be automatic. Manual tuning is often time consuming and can lead to sub-optimal results, for example, by introducing human error. Further, automatic tuning or “autotuning” is frequently inaccurate and/or unreliable. If a process disturbance occurs during such autotuning, the controller can erroneously set the PI parameters as if the transitory disturbance was part of the system cause-effect response. Further, system models, often used to estimate the tuning parameters from information measured in the system, can be difficult to identify online.

What is needed are systems and methods for autotuning an HVAC system controller.

SUMMARY

Embodiments of the disclosure may provide an exemplary method for autotuning an HVAC system controller. The method may include receiving a performance coefficient, and providing a step-input using an autotuner relay to measure an ultimate gain and an ultimate period in a controlled HVAC system. The method may also include adjusting the ultimate gain and the ultimate period to account for a relay hysteresis, and applying a tuning rule to derive a proportional gain and an integral time for a controller of the HVAC system control loop. The method may further include updating the controller with the proportional gain and integral time, and verifying the proportional gain and integral time. Verifying the proportional gain and integral time may include setting a performance envelope having a tightness related to the performance coefficient, applying a step-input to provoke a closed-loop response, and comparing the closed loop response with the performance envelope to determine whether the closed-loop response is within the envelope.

Embodiments of the disclosure may provide an exemplary HVAC system. The HVAC system may include a fan coil unit including a fan and a heat exchange coil, a duct extending between the fan coil unit and a volume of air, and a sensor disposed in or proximal to the volume of air and configured to sense the air temperature of a certain zone. The HVAC system may also include a controller coupled to the fan coil unit and the sensor and configured to receive a signal from the sensor and provide a signal to the fan coil unit so as to regulate the air temperature of a certain zone. The controller may include an input device that is selectable to provide a performance coefficient to the controller so as to provide a performance envelope to verify a controller autotuning result.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate aspects of the present teachings and together with the description, serve to explain principles of the present teachings. In the figures:

<figref idref="DRAWINGS">FIG. 1</figref> illustrates a schematic view of an exemplary HVAC system, according to an embodiment.

<figref idref="DRAWINGS">FIG. 2</figref> illustrates an exemplary block diagram of a control feedback loop with a PI controller, according to an embodiment.

<figref idref="DRAWINGS">FIG. 3</figref> illustrates a flowchart of an exemplary method for autotuning a controller, according to an embodiment.

<figref idref="DRAWINGS">FIG. 4</figref> illustrates a flowchart of an exemplary embodiment of calculating PI parameters in an embodiment of the method for autotuning the controller.

<figref idref="DRAWINGS">FIG. 5</figref> illustrates a plot of a relay with hysteresis for autotuning, according to an embodiment.

<figref idref="DRAWINGS">FIG. 6</figref> illustrates a plot of adjustment factors for ultimate gain and ultimate period as a function of a ratio between the time delay (D) and the time constant (T) of the controlled system, according to an embodiment.

<figref idref="DRAWINGS">FIG. 7</figref> illustrates a plot of proportional gain derived from a tuning rule as compared to internal model control based results, according to an embodiment.

<figref idref="DRAWINGS">FIG. 8</figref> illustrates a plot of integral time derived from a tuning rule as compared to internal model control based results, according to an embodiment.

<figref idref="DRAWINGS">FIG. 9</figref> illustrates a flowchart of an exemplary embodiment of verifying the tuning results in an embodiment of the method for autotuning the controller, according to an embodiment.

<figref idref="DRAWINGS">FIG. 10</figref> illustrates a plot of a closed loop response in a performance envelope, for verifying the tuning results, according to an embodiment.

<figref idref="DRAWINGS">FIG. 11</figref> illustrates a plot of the ratio of natural frequency to ultimate frequency versus D/T for three tuning settings, according to an embodiment.

<figref idref="DRAWINGS">FIG. 12</figref> illustrates a plot of closed-loop damping ratio versus D/T for three tuning settings, according to an embodiment.

DETAILED DESCRIPTION

Reference will now be made in detail to various embodiments of the present teachings, an example of which is illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts.

In the following description, reference is made to the accompanying drawings that form a part thereof, and in which is shown by way of illustration specific implementations in which may be practiced. These implementations are described in sufficient detail to enable those skilled in the art to practice these implementations and it is to be understood that other implementations may be utilized and that changes may be made without departing from the scope of the present teachings. The following description is, therefore, merely exemplary.

<figref idref="DRAWINGS">FIG. 1</figref> illustrates a schematic view of an exemplary HVAC system <b>100</b>, according to an embodiment. The HVAC system <b>100</b> is configured to chill and/or heat air supplied to a volume or “zone” <b>101</b>, which may be a facility, room, or portion thereof. In an exemplary embodiment, the HVAC system <b>100</b> includes a fan coil unit (FCU) <b>102</b>, a controller <b>104</b>, and a sensor <b>106</b>. It will be appreciated that a variety of other HVAC systems <b>100</b> may be employed, without departing from the scope of the present disclosure. Further, in the illustrated embodiment, the FCU <b>102</b> may include a fan <b>108</b> and a coil <b>110</b>. The fan <b>108</b> may be any suitable axial, radial, or centrifugal fan or blower, and the coil <b>110</b> may be any suitable heat exchanger, such as a heat exchange coil. Further, the coil <b>110</b> may form part of a heat pump or other refrigerant cycle so as to heat or cool air pushed across the coil <b>110</b> by the fan <b>108</b>. The FCU <b>102</b> may be configured to supply air to the zone <b>101</b> via a duct <b>112</b>, which may be any suitable size and shape and may have one or more inlets and/or one or more outlets, so as to distribute air flow in the zone <b>101</b> and/or to direct air to other zones (not shown).

The sensor <b>106</b> may be disposed in or proximal to the zone <b>101</b> and may be a temperature sensor such as a thermometer, thermistor, or thermocouple, or any other type of sensor. The sensor <b>106</b> may thus be configured to sense one or more relevant conditions in the zone <b>101</b>. The sensor <b>106</b> may be coupled to the controller <b>104</b> and configured to provide signals thereto indicative of the relevant condition(s) in the zone <b>101</b>.

The controller <b>104</b> may, in turn, be coupled to the FCU <b>102</b> and configured to control the speed of the fan <b>108</b>, the flow rate of heat transfer medium (e.g., water) through the coil <b>110</b>, other variables, or a combination thereof. The controller <b>104</b> may be or include one or more proportional-integral (PI) controllers, proportional-integral-derivative (PID) controllers, and/or the like. Accordingly, the controller <b>104</b> may receive a signal from the sensor <b>106</b> and respond by adjusting the operating parameters of the FCU <b>102</b> to achieve a desired setpoint in the zone <b>101</b>.

Further, the controller <b>104</b> may have one, two, or more input devices, for example, at least a first input device <b>114</b> and a second input device <b>116</b>, as shown. The first input device <b>114</b> may be, at least conceptually, a switch, configured to signal to the controller <b>104</b> that autotuning is desired. Accordingly, the first input device <b>114</b> may be a manual switch, toggled by an operator, or may be a time delay switch, configured to start the autotuning process at set intervals. The second input device <b>116</b> may be a knob, dial, digital input, or may instead be a constant, pre-determined according to end-user needs or other factors. The second input device <b>116</b> may provide an acceptable performance coefficient α to the controller <b>104</b>, which may be employed during autotuning to set an acceptable performance envelope “tightness,” as will be described in greater detail below.

<figref idref="DRAWINGS">FIG. 2</figref> illustrates a control block diagram of an exemplary control loop <b>200</b> for an HVAC system, such as the HVAC system <b>100</b>, described above. The control loop <b>200</b> may include a setpoint input Ysp, which is fed to a signal comparator <b>202</b>, where the setpoint input Ysp is compared to a feedback signal via line <b>204</b>. The result, an “error” signal, is indicative of the difference between the setpoint input Ysp and the feedback signal via line <b>204</b>. The error signal is then fed to a controller <b>206</b>, which applies a control function C(s) to derive an output U. The controller <b>206</b> may be a PI controller, a PID controller, a combination thereof, or the like and may form part of or include the controller <b>104</b> (<figref idref="DRAWINGS">FIG. 1</figref>), for example.

The output U is then fed to a component or “controlled system” <b>208</b>, which may be or include the FCU <b>102</b> and/or another component. The controlled system <b>208</b> reacts to the output U and generates an output signal Y. The output signal Y is then fed back to the signal comparator <b>202</b> via the line <b>204</b>.

The control loop <b>200</b> may also include an autotuner relay <b>210</b> disposed in parallel with the controller <b>206</b>. The autotuner relay <b>210</b> may be selectively engaged, for example, when a signal to start the autotuning is received from the first input device <b>114</b> (<figref idref="DRAWINGS">FIG. 1</figref>). When engaged, the autotuner relay <b>210</b> may take over control of the control loop <b>200</b> from the controller <b>206</b> and provide a periodic step function to the controlled system <b>208</b>. By application of the step function, the autotuner relay <b>210</b> may determine a limit cycle of the controlled system <b>208</b>, and thus the ultimate gain Ku and the ultimate period Pu thereof. As will be more fully described below, with knowledge of the ultimate gain Ku and ultimate period Pu, and any adjustment as necessary, for example, as provided herein and described below, the PI parameters of proportional gain Kc and integral time τ can be determined, so as to tune the controller <b>206</b>.

With additional reference to <figref idref="DRAWINGS">FIGS. 1 and 2</figref>, <figref idref="DRAWINGS">FIG. 3</figref> illustrates an exemplary method <b>300</b> for autotuning a controller, i.e., determining and setting PI parameters, according to an embodiment. In an exemplary embodiment, the method <b>300</b> may be used to estimate a point on a Nyquist plot of the controlled system <b>208</b> where the phase lag is −π.

The method <b>300</b> may begin by setting the performance coefficient α, as at <b>302</b>. The performance coefficient α may be used to determine the maximum overshoot Mp and acceptable damping ratio ζ, as will be described in greater detail below. Setting the performance coefficient α at <b>302</b> may include receiving the coefficient α as an input from a user (e.g., via the second input device <b>116</b>, shown in <figref idref="DRAWINGS">FIG. 1</figref>), for example, during maintenance, at scheduled intervals, or may be coded or wired into the controller <b>104</b> as a constant.

The method <b>300</b> may then proceed to checking to see if the controlled system <b>208</b> is presently operating at or close to steady state, as at <b>304</b>. One way to perform such checking at <b>304</b> may be to compare a previous number of outputs Y and U, to determine if they are within a certain range. For example, the previous about 100, 1000, 10000, or more outputs Y and U may be checked.

The method <b>300</b> may then proceed to switching to relay control, as at <b>306</b>. Switching the control loop <b>200</b> of the HVAC system <b>100</b> to relay control at <b>306</b> may cause the autotuner relay <b>210</b> to take over from the controller <b>206</b> in the control loop <b>200</b>. The method <b>300</b> may then proceed to calculating the PI parameters, e.g., proportional gain Kc and integral time τ, as at <b>308</b> by relay feedback autotuning.

Referring now additionally to <figref idref="DRAWINGS">FIG. 4</figref>, there is shown a flowchart of an exemplary embodiment of calculating the PI parameters, as at <b>308</b>, of an embodiment of the method <b>300</b>. For example, calculating the PI parameters at <b>308</b> may include continuing the application of the relay control, originated during the switching at <b>306</b> (<figref idref="DRAWINGS">FIG. 3</figref>), so as to calculate an ultimate gain Ku and ultimate period Pu of the controlled system <b>208</b>, as at <b>312</b>. The dynamics of control loops for a wide range of HVAC systems may be approximated according to a first order plus dead time (FOPDT), whereby:

<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>=</mo><mfrac><msup><mi>Kⅇ</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup><mrow><mi>Ts</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

In equation (1), K is the process gain, T is the time constant, and D is the time delay. For HVAC system <b>100</b> components such as the zone <b>101</b> (<figref idref="DRAWINGS">FIG. 1</figref>), the ratio of the system time delay to the time constant (D/T) may be assumed to be between about 0.1 and about 0.5; however, in some situations, an acceptable D/T ratio may span a wider range, for example, between about 0.1 and about 1 or greater. It will be appreciated that as D/T ratios exceed about 1, tuning may have an attenuated impact in some systems.

The application of a relay step function may produce sustained limit cycles in the controlled system <b>208</b>. Such sustained limit cycles may be considered present when the periods of the latest two limit cycles are within about 30% of each other. As an illustrative example, the step function may initially have a positive output change with magnitude h, until the controlled system <b>208</b> response crosses its setpoint. At this time, the autotuner relay <b>210</b> may reverse (or reduce or zero) its output change with magnitude −h, until the controlled system <b>208</b> output Y comes back and crosses its setpoint again. Repetition of this step function application can generate limit cycles, in which the ultimate frequency ω_uis the frequency of the oscillation of the controlled system <b>208</b>, as just described, while the ultimate gain is:

<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>K</mi><mi>u</mi></msub><mo>=</mo><mfrac><mrow><mn>4</mn><mo>⁢</mo><mi>h</mi></mrow><mrow><mi>π</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>a</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

In equation (2), a is the magnitude of oscillation of the controlled system <b>208</b>, and h is the amplitude of the step function applied by the autotuner relay <b>210</b>. In various embodiments consistent with use in HVAC systems, h may be between about 5% and about 10% of the control output scale. Further, the control error ε may be set at between about 2 and about 5 times the measurement of the noise covariance, which may be determined in any suitable way.

Calculating the PI parameters at <b>308</b> may also include applying relay hysteresis compensation, as at <b>314</b>. Relay hysteresis may be employed to prevent the autotuner relay <b>210</b> from switching due to control error signal noise. <figref idref="DRAWINGS">FIG. 5</figref> illustrates the effect of relay hysteresis, whereby, in the top horizontal line, the control output moves from Uo (the steady state output of the controller <b>206</b>) plus the amplitude h of the step function, to Uo-h, as the control error proceeds to −ε. However, such hysteresis may introduce error to the determination of the ultimate period Pu and ultimate gain Ku. For example, the ultimate gain Ku may be smaller than that of an ideal relay feedback.

Accordingly, calculating the PI parameters at <b>308</b> may proceed to adjusting the ultimate gain Ku and the ultimate period Pu, as at <b>316</b>, to account for the relay hysteresis impact. Generally, the adjustment of the ultimate gain may proceed according to the following equation. Reference may be made to paper by Astrom and Hagglund, <i>Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins</i>, AUTOMATICA, 20, 645-651 (1984), the entirety of which is incorporated herein by reference to the extent consistent with the present disclosure.

<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mrow><msubsup><mi>K</mi><mi>u</mi><mi>′</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><mn>4</mn><mo>⁢</mo><mi>h</mi></mrow><mrow><mi>π</mi><mo>⁢</mo><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>ɛ</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>=</mo><mrow><mfrac><mn>1</mn><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mrow><mo>(</mo><mrow><mi>ɛ</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>a</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mfrac><mo>⁢</mo><msub><mi>K</mi><mi>u</mi></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Furthermore, the ultimate period Pu (and thus, the ultimate frequency) may also be altered. For the FOPDT transfer function G(s) of the controlled system <b>208</b>, the adjustment to the ultimate gain Ku and ultimate period Pu may be modeled according to the following functions:

<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mrow><msubsup><mi>K</mi><mi>u</mi><mi>′</mi></msubsup><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mrow><mrow><mo>(</mo><mrow><mi>ɛ</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>a</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mrow><mo>-</mo><mi>D</mi></mrow><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>T</mi></mrow></msup></mrow></mrow></mfrac><mo>⁢</mo><msub><mi>K</mi><mi>u</mi></msub></mrow></mrow></math></maths><maths id="MATH-US-00004-2" num="00004.2"><math overflow="scroll"><mrow><msubsup><mi>P</mi><mi>u</mi><mi>′</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>ln</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo>⁢</mo><msup><mi>e</mi><mrow><mi>D</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>T</mi></mrow></msup></mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>ln</mi><mo>⁢</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><msup><mi>e</mi><mrow><mi>D</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>T</mi></mrow></msup></mrow><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>-</mo><mrow><mi>s</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>a</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mrow><mi>ɛ</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>a</mi></mrow></mrow></mfrac></mrow></mfrac><mo>-</mo><msub><mi>P</mi><mi>u</mi></msub></mrow></mrow></math></maths>

<figref idref="DRAWINGS">FIG. 6</figref> illustrates a plot of the adjusting factor (K′u/Ku and P′u/Pu from equations (4) and (5) above) as a function of ε/a and D/T, according to an exemplary embodiment. As can be appreciated, with a larger ε/a and smaller D/T, the adjustment factor may increase. However, since the controlled system <b>208</b> may be an HVAC system and/or component thereof and may be analytically known to have a D/T of between about 0.1 and about 0.5, the average adjusting factors may be employed to provide the adjusted ultimate gain K′u and the adjusted ultimate period P′u, without knowing the precise D/T value.

Calculating the PI parameters at <b>308</b> may further proceed to applying a tuning rule, such as Ziegler-Nichols (Z-N) to derive the proportional gain Kc and integral time τ, as at <b>318</b>, from the adjusted ultimate gain K′u and ultimate period P′u. It will be readily appreciated that other tuning rules are within the scope of the present disclosure, such as Tyreys-Luyben.

According to Z-N, the proportional gain Kc and integral time τ may be readily calculated from the following equations: &lt;br /&gt;<i>Kc=Ku/</i>2.2  (6)&lt;br /&gt;τ=<i>Pu/</i>1.2  (7)

Calculating the PI parameters at <b>308</b> may then proceed to adjusting the proportional gain Kc and the integral time τ using internal model control (IMC) results. For a known process model (e.g., such as an HVAC system), tuning parameters can be derived for a wide variety of transfer function models. One such IMC-based tuning formula is Skogestad IMC-based tuning, where the PI controller transfer function is &lt;br /&gt;<i>C</i>(<i>s</i>)=<i>Kc</i>(1+1/τ<i>s</i>)  (8)

Skogestad's IMC-based PI tuning rule for a FOPDT system is: &lt;br /&gt;<i>Kc=T/</i>2<i>KD,t</i>=min(<i>T,</i>8<i>D</i>)  (9)

One embodiment of Skogestad's IMC-based tuning is described in Skogestad, <i>Simple Analytic Rules for Model Reduction and PID Controller Tuning</i>, J. <smallcaps>OF </smallcaps>P<smallcaps>ROCESS </smallcaps>C<smallcaps>ONTROL, </smallcaps>13, 291-309 (2003), the entirety of which is incorporated herein by reference, to the extent consistent with the present disclosure. Comparison of the tuning results from applying the tuning rule at <b>318</b>, with those derived from the IMC results may provide good setpoint tracking and disturbance rejection.

<figref idref="DRAWINGS">FIGS. 7 and 8</figref> illustrate an exemplary plot of proportional gain Kc as a function of D/T and integral time τ as a function of D/T, as derived from the tuning rule at <b>318</b> and from the IMC-based tuning. As shown, where D/T is between about 0.1 and about 0.5, the Z-N tuning rule derived proportional gain Kc is about 76.5% higher on average, while the Z-N tuning rule derived τ is lower when D/T is less than about 0.34, but higher when D/T is greater than about 0.34. Accordingly, when D/T is between about 0.1 and about 0.5, a range for HVAC systems, relay feedback using the tuning results, as at <b>318</b>, may be compared and adjusted to yield approximately the same results as the IMC-based tuning. As such, the Z-N tuning rule for proportional gain may be adjusted to: &lt;br /&gt;<i>Kc=Ku/</i>3.9  (10)

Furthermore, as can be appreciated from <figref idref="DRAWINGS">FIG. 7</figref>, 1.2Pu/T is approximately linear in relationship to D/T. Accordingly, for FOPDT with D/T being between about 0.1 and about 0.5, the time delay D may be estimated as &lt;br /&gt;D≈0.3Pu  (11)

Referring again to <figref idref="DRAWINGS">FIG. 3</figref>, the method <b>300</b> may proceed to pre-updating the PI tuning, as at <b>322</b>. This may include providing the controller <b>206</b> with the newly-calculated proportional gain Kc and integral time τ. The method <b>300</b> may then proceed to switching back to PI control, as at <b>324</b>, for example, by bumpless transition. Additionally, the method <b>300</b> may include verifying the tuning results, as at <b>326</b>.

Referring now additionally to <figref idref="DRAWINGS">FIG. 9</figref>, a flowchart of an exemplary embodiment of verifying the tuning results at <b>326</b> is illustrated. For example, verifying the tuning results at <b>326</b> may include setting an envelope, as at <b>328</b>, providing a step-input to test the closed loop response of the newly-tuned control, as at <b>330</b>, and determining whether the tuning results in the closed-loop response that is in the envelope, as at <b>332</b>.

In various systems, the method <b>300</b> may be impacted by external deterministic disturbances during execution. Accordingly, verifying the obtained tuning parameters may be desired to avoid inaccurate tuning. The method <b>300</b> may thus proceed to setting a performance envelope at <b>328</b>. The use of a performance envelope may be used in addition to or in lieu of comparing the response characteristics (e.g., decay ratio, settling time, overshoot, etc.) or a combination thereof to a desired value.

Setting the performance envelope at <b>328</b> may include deriving boundaries for an acceptable closed-loop response. The closed-loop transfer functions are:

<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>Y</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><mrow><mi>C</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mi>C</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><msub><mi>Y</mi><mi>sp</mi></msub></mrow><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>A</mi><mo>=</mo><mn>0</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>12</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>U</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mrow><mrow><mi>C</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mrow><mrow><mi>C</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>G</mi><mo>⁡</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><mrow><mo>(</mo><mrow><mo>-</mo><mi>A</mi></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>No</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>Ysp</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>change</mi></mrow><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

It will be appreciated from equations (12) and (13) that the closed-loop transfer function is the same between Y and Ysp and U and (−A); therefore, the method <b>300</b> may include observing output U response to assess the control tuning results. To observe the closed loop output U response, a step disturbance A may be applied, as at <b>330</b>, for example, by adding a bias to the controller integral term, i.e., adding the step disturbance A between the controller <b>206</b> and the controlled system <b>208</b> in <figref idref="DRAWINGS">FIG. 2</figref>. This perturbation may allow the controlled system <b>208</b> to return to its previous steady state after the test.

<figref idref="DRAWINGS">FIG. 10</figref> illustrates an exemplary control response and performance window or “envelope” <b>1001</b> of acceptable closed-loop responses <b>1002</b>, according to an embodiment. The envelope may be established using a maximum acceptable overshoot Mp, which may link gain margin and phase margin through the damping ratio. The input and calculation of the maximum acceptable overshoot will be described below.

Further, the envelope may be defined by several time points, for example, time points t₁, t₂, t₃, t₄, as shown in <figref idref="DRAWINGS">FIG. 10</figref>. The first time point t₁may be a time that is reasonably close to, but greater than, the apparent time delay D. For example, the first time point t₁may be about 1.5 D. The second time point t₂may be set such that maximum overshoot occurs between the first and second time points t₁and t₂; accordingly, if the overshoot is higher than acceptable, the response may be outside of the envelope between the first and second time points t₁and t₂. As such, the second time point t₂may be the time delay plus peak time, i.e.:

<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mrow><mi>D</mi><mo>+</mo><mfrac><mi>π</mi><mrow><msub><mi>ω</mi><mi>n</mi></msub><mo>⁢</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

In equation (14), the ω_nis the natural frequency ζ and is the damping ratio. Estimation of ω_nand ζ will be discussed below.

The third time point t₃may be the time constant of an exponential term that determines how sluggishly the closed-loop response moves toward its steady state. The desired time constant may be D, based on the tuning target; however, to detect an overly-sluggish response, the time point t₃may be set to about 3 D. The fourth time point t₄sets a boundary to detect an under-damped case. If the closed loop response is under-damped, the transient response along the overshoot side may be bounded by a curve which has an exponential term determining how fast the response is required to move to its steady state. Further, the fifth time point t₅may provide response monitoring time, which may be about equal to D+3(t₃) for over-damped response and D+3/ω_nζ for under-damped response.

To determine the second time point t₂from the equation (14) above, the natural frequency must be determined Since the natural frequency ω_n, of a closed-loop response depends on both the controlled system and the control tuning, it may be estimated. Referring to equations (1) and (8), the closed-loop transfer function between A and U may be:

<maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mi>U</mi><mi>A</mi></mfrac><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mrow><msub><mi>KK</mi><mi>C</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>τ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>s</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup></mrow><mrow><mrow><mi>τ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><mi>s</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>Ts</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mrow><msub><mi>KK</mi><mi>C</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mrow><mi>τ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>s</mi></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup></mrow></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

For relay feedback autotuning, the proportional gain Kc may be determined by equation (10); however, the integral time τ may be greater or smaller than the process time constant T, depending on the D/T ratio. As such, the tuned integral time τ may be a too low or too high as compared to IMC tuning for τ=T, where D/T is greater than or equal to about 0.125 or τ=8 D for D/T is less than about 0.125. Accordingly, to estimate the fourth time point t₄, the effect of the tuning impact on the natural frequency may be determined, employing three integral time tunings, τ=∞ (no integral action), τ=T (IMC tuning), and τ=D (strong integral time).

In the first case, where τ=∞, a first-order Pade approximation is used to determine the time delay in the denominator of equation (15), yielding:

<maths id="MATH-US-00008" num="00008"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mi>U</mi><mi>A</mi></mfrac><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mfrac><msub><mi>KK</mi><mi>C</mi></msub><mi>TD</mi></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mi>Ds</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup></mrow><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mrow><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi></mrow><mo>+</mo><mi>D</mi><mo>-</mo><mrow><msub><mi>KK</mi><mi>C</mi></msub><mo>⁢</mo><mi>D</mi></mrow></mrow><mi>TD</mi></mfrac><mo>⁢</mo><mi>s</mi></mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>C</mi></msub></mrow><mo>)</mo></mrow></mrow><mi>TD</mi></mfrac></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Therefore, for 0&lt;ζ&lt;1, the natural frequency ω_nand damping ratio ζ are:

<maths id="MATH-US-00009" num="00009"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>ω</mi><mi>n</mi></msub><mo>=</mo><msqrt><mfrac><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>C</mi></msub></mrow><mo>)</mo></mrow></mrow><mi>TD</mi></mfrac></msqrt></mrow><mo>,</mo><mrow><mrow><mi>and</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>ζ</mi></mrow><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi></mrow><mo>+</mo><mi>D</mi><mo>-</mo><mrow><msub><mi>KK</mi><mi>C</mi></msub><mo>⁢</mo><mi>D</mi></mrow></mrow><mrow><mn>2</mn><mo>⁢</mo><msqrt><mrow><mn>2</mn><mo>⁢</mo><mrow><mi>TD</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>C</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msqrt></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>17</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Furthermore, the ultimate frequency ω_umay be obtained by setting Kc=Ku, which yields:

<maths id="MATH-US-00010" num="00010"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><msub><mi>ω</mi><mi>n</mi></msub><msub><mi>ω</mi><mi>u</mi></msub></mfrac><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi>KK</mi><mi>u</mi></msub><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mn>3.9</mn></mrow></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>u</mi></msub></mrow></mfrac></msqrt></mrow></mtd><mtd><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

In the second case, where τ=T, equation (15) resolves to:

<maths id="MATH-US-00011" num="00011"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mi>U</mi><mi>A</mi></mfrac><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mfrac><msub><mi>KK</mi><mi>C</mi></msub><mi>TD</mi></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mi>Ds</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup></mrow><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mrow><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi></mrow><mo>-</mo><mrow><msub><mi>KK</mi><mi>C</mi></msub><mo>⁢</mo><mi>D</mi></mrow></mrow><mi>TD</mi></mfrac><mo>⁢</mo><mi>s</mi></mrow><mo>+</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>KK</mi><mi>C</mi></msub></mrow><mi>TD</mi></mfrac></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>19</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

As shown in Table 1, below, the formula for ω_n/ω_uand damping ratio ζ may be known for different cases. Further, in the third case shown below, i.e., where τ=D, first order dynamics may be used to approximate the time delay term in the denominator of equation (15), yielding:

<maths id="MATH-US-00012" num="00012"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mi>U</mi><mi>A</mi></mfrac><mo>=</mo><mrow><mo>-</mo><mfrac><mrow><mfrac><msub><mi>KK</mi><mi>C</mi></msub><mi>TD</mi></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>Ds</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mi>Ds</mi></mrow></msup></mrow><mrow><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>⁢</mo><mi>s</mi></mrow><mo>+</mo><mfrac><msub><mi>KK</mi><mi>C</mi></msub><mi>TD</mi></mfrac></mrow></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

The formula for ω_n/ω_uand ζ of this case is also shown in Table 1.

<tables id="TABLE-US-00001" num="00001"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="3"><colspec colname="1" colwidth="70pt" align="left" /><colspec colname="2" colwidth="63pt" align="center" /><colspec colname="3" colwidth="84pt" align="center" /><thead><row><entry namest="1" nameend="3" align="center" rowsep="1" /></row><row><entry>Tuning Parameters</entry><entry>ω_n/ω_u</entry><entry>Damping Ratio ζ </entry></row><row><entry namest="1" nameend="3" align="center" rowsep="1" /></row></thead><tbody valign="top"><row><entry>Kc = Ku/3.9</entry><entry><maths id="MATH-US-00013" num="00013"><math overflow="scroll"><msqrt><mfrac><mrow><mn>1</mn><mo>+</mo><mrow><msub><mi>KK</mi><mi>u</mi></msub><mo>/</mo><mn>3.9</mn></mrow></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>u</mi></msub></mrow></mfrac></msqrt></math></maths></entry><entry><maths id="MATH-US-00014" num="00014"><math overflow="scroll"><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi></mrow><mo>+</mo><mi>D</mi><mo>-</mo><msub><mi>DKK</mi><mi>C</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><msqrt><mrow><mn>2</mn><mo>⁢</mo><mrow><mi>TD</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>C</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msqrt></mrow></mfrac></math></maths></entry></row><row><entry></entry></row><row><entry>Kc = Ku/3.9, τ = T</entry><entry><maths id="MATH-US-00015" num="00015"><math overflow="scroll"><msqrt><mfrac><mrow><msub><mi>KK</mi><mi>u</mi></msub><mo>/</mo><mn>3.9</mn></mrow><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>u</mi></msub></mrow></mfrac></msqrt></math></maths></entry><entry><maths id="MATH-US-00016" num="00016"><math overflow="scroll"><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi></mrow><mo>-</mo><msub><mi>DKK</mi><mi>C</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><msqrt><mrow><mn>2</mn><mo>⁢</mo><msub><mi>TDKK</mi><mi>C</mi></msub></mrow></msqrt></mrow></mfrac></math></maths></entry></row><row><entry></entry></row><row><entry>Kc = Ku/3.9, τ = D</entry><entry><maths id="MATH-US-00017" num="00017"><math overflow="scroll"><msqrt><mfrac><mrow><msub><mi>KK</mi><mi>u</mi></msub><mo>/</mo><mn>3.9</mn></mrow><mrow><mn>2</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>KK</mi><mi>u</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></msqrt></math></maths></entry><entry><maths id="MATH-US-00018" num="00018"><math overflow="scroll"><mfrac><msqrt><mrow><mi>D</mi><mo>/</mo><msub><mi>TKK</mi><mi>C</mi></msub></mrow></msqrt><mn>2</mn></mfrac></math></maths></entry></row><row><entry namest="1" nameend="3" align="center" rowsep="1" /></row></tbody></tgroup></table></tables>

Considering a step input disturbance A with amplitude h and 0&lt;ζ&lt;1, the time domain response of equation (19) becomes:

<maths id="MATH-US-00019" num="00019"><math overflow="scroll"><mtable><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mrow><mi>u</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mi>D</mi></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>U</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mi>D</mi></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi>A</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mi>D</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mi>h</mi><mo>,</mo></mrow></mtd><mtd><mrow><mn>0</mn><mo>≤</mo><mrow><mi>t</mi><mo>+</mo><mi>D</mi></mrow><mo>≤</mo><mi>D</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>h</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msup><mi>e</mi><mrow><mo>-</mo><msub><mi>ζω</mi><msup><mi>a</mi><mi>′</mi></msup></msub></mrow></msup><mo>·</mo><msqrt><mfrac><mrow><mrow><msup><mi>T</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ω</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>cos</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ϕ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mo>·</mo></mrow></mrow></mtd><mtd><mrow><mrow><mi>t</mi><mo>+</mo><mi>D</mi></mrow><mo>&gt;</mo><mi>D</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>sin</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>ω</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>t</mi><mo>⁢</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>-</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable><mo>⁢</mo><mstyle><mtext></mtext></mstyle><mo>⁢</mo><mrow><mrow><mi>where</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>ϕ</mi></mrow><mo>=</mo><mrow><mrow><msup><mi>tg</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>⁢</mo><mfrac><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></msqrt><mi>ζ</mi></mfrac><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>and</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>θ</mi></mrow><mo>=</mo><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>⁢</mo><mfrac><mrow><mi>sin</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ϕ</mi></mrow><msqrt><mrow><mrow><msup><mi>T</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ω</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>cos</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ϕ</mi></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mfrac></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>21</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Similar to a standard second-order system response, equation (21) shows that the overshoot side is bounded by a curve, defined as:

<maths id="MATH-US-00020" num="00020"><math overflow="scroll"><mrow><mi>h</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mrow><msup><mi>e</mi><mrow><mrow><mo>-</mo><msub><mi>ζω</mi><mi>a</mi></msub></mrow><mo>⁢</mo><mi>t</mi></mrow></msup><mo>·</mo><msqrt><mfrac><mrow><mrow><msup><mi>T</mi><mn>2</mn></msup><mo>⁢</mo><msup><mi>ω</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><mi>T</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ω</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>cos</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>ϕ</mi></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></mfrac></msqrt></mrow></mrow></math></maths>

Further, the fourth time point t₄=1/ω_nζ.

<figref idref="DRAWINGS">FIGS. 11 and 12</figref> illustrate exemplary plots of ω_n/ω_uand ζ for the three integral time settings discussed above, as a function of D/T between 0.1 and about 0.5, and employing the FOPDT approximation. The average ω_n/ω_uis thus employed for estimating the fourth time point t4. As illustrated in the <figref idref="DRAWINGS">FIG. 11</figref>, the average ω_n/ω_uis about 0.468. Using equation (11) yields:

<maths id="MATH-US-00021" num="00021"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>t</mi><mn>4</mn></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>ω</mi><mi>n</mi></msub><mo>⁢</mo><mi>ζ</mi></mrow></mfrac><mo>=</mo><mrow><mfrac><msub><mi>P</mi><mi>u</mi></msub><mrow><mn>0.936</mn><mo>⁢</mo><mi>πζ</mi></mrow></mfrac><mo>≈</mo><mfrac><mrow><mn>1.134</mn><mo>⁢</mo><mi>D</mi></mrow><mi>ζ</mi></mfrac></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>22</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

The maximum acceptable overshoot Mp is related to the damping ratio ζ, such that

<maths id="MATH-US-00022" num="00022"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>M</mi><mi>p</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mrow><mo>-</mo><mi>ζπ</mi></mrow><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></msqrt></mrow></msup></mrow></mtd><mtd><mrow><mo>(</mo><mn>23</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Accordingly, the maximum acceptable overshoot Mp setting may be employed to determine the damping ratio for the envelope determination. Such setting, as at <b>302</b>, may proceed by the user turning a knob, or adjusting a dial, etc., such that a performance coefficient α is determined, such that: &lt;br /&gt;<i>Mp=</i>0.5−0.35α  (24)&lt;br /&gt;<i>t</i>₃=5<i>D−</i>3.5α<i>D</i>  (25)

Where a lower coefficient α corresponds to a looser envelope and a higher coefficient α corresponds to a tighter envelope; accordingly, the performance coefficient α is employed to determine the “tightness” of the performance envelope. The parameters for the envelope may be set as:

<maths id="MATH-US-00023" num="00023"><math overflow="scroll"><mrow><mrow><mi>Mp</mi><mo>=</mo><mrow><mn>0.5</mn><mo>-</mo><mrow><mn>0.35</mn><mo>⁢</mo><mi>α</mi></mrow></mrow></mrow><mo>,</mo><mrow><mi>with</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>α</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>being</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>an</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>input</mi></mrow><mo>,</mo><mrow><mi>which</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>is</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>used</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>to</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>derive</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>the</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>corresponding</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mi>ζ</mi></mrow></mrow></math></maths><maths id="MATH-US-00023-2" num="00023.2"><math overflow="scroll"><mrow><mstyle><mspace width="4.4em" height="4.4ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mrow><mn>1.5</mn><mo>⁢</mo><mi>D</mi></mrow></mrow></mrow></math></maths><maths id="MATH-US-00023-3" num="00023.3"><math overflow="scroll"><mrow><mstyle><mspace width="4.4em" height="4.4ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mrow><mrow><mi>D</mi><mo>+</mo><mfrac><mi>π</mi><mrow><msub><mi>ω</mi><mi>n</mi></msub><mo>⁢</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msup><mi>ζ</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow><mo>≈</mo><mrow><mn>4.6</mn><mo>⁢</mo><mi>D</mi></mrow></mrow></mrow></mrow></math></maths><maths id="MATH-US-00023-4" num="00023.4"><math overflow="scroll"><mrow><mstyle><mspace width="4.4em" height="4.4ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mrow><mrow><mn>5</mn><mo>⁢</mo><mi>D</mi></mrow><mo>-</mo><mrow><mn>3.5</mn><mo>⁢</mo><mi>α</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>D</mi></mrow></mrow></mrow></mrow></math></maths><maths id="MATH-US-00023-5" num="00023.5"><math overflow="scroll"><mrow><mstyle><mspace width="4.4em" height="4.4ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>t</mi><mn>4</mn></msub><mo>=</mo><mrow><mn>1.134</mn><mo>⁢</mo><mi>D</mi><mo>⁢</mo><mstyle><mtext>/</mtext></mstyle><mo>⁢</mo><mi>ζ</mi></mrow></mrow></mrow></math></maths><maths id="MATH-US-00023-6" num="00023.6"><math overflow="scroll"><mrow><mstyle><mspace width="4.4em" height="4.4ex" /></mstyle><mo>⁢</mo><mrow><msub><mi>t</mi><mn>5</mn></msub><mo>=</mo><mrow><mi>min</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><mrow><mo>{</mo><mrow><mrow><mi>D</mi><mo>+</mo><mrow><mn>3</mn><mo>⁢</mo><msub><mi>t</mi><mn>3</mn></msub></mrow></mrow><mo>,</mo><mrow><mi>D</mi><mo>+</mo><mrow><mn>3</mn><mo>⁢</mo><msub><mi>t</mi><mn>4</mn></msub></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths>

Accordingly, with knowledge of each of the variables, and by input of the performance coefficient α, the performance envelope can be set. The closed loop system response can then be compared to the envelope, to determine if the tuning results are acceptable, thereby verifying the results, as at <b>326</b>.

While the present teachings have been illustrated with respect to one or more implementations, alterations and/or modifications can be made to the illustrated examples without departing from the spirit and scope of the appended claims. For example, it will be appreciated that while the process is described as a series of acts or events, the present teachings are not limited by the ordering of such acts or events. Some acts may occur in different orders and/or concurrently with other acts or events apart from those described herein. Also, not all process stages may be required to implement a methodology in accordance with one or more aspects or embodiments of the present teachings.

It will be appreciated that structural components and/or processing stages can be added or existing structural components and/or processing stages can be removed or modified. Further, one or more of the acts depicted herein may be carried out in one or more separate acts and/or phases. Furthermore, to the extent that the terms “including,” “includes,” “having,” “has,” “with,” or variants thereof are used in either the detailed description and the claims, such terms are intended to be inclusive in a manner similar to the term “comprising.” The term “at least one of” is used to mean one or more of the listed items can be selected. Further, in the discussion and claims herein, the term “on” used with respect to two materials, one “on” the other, means at least some contact between the materials, while “over” means the materials are in proximity, but possibly with one or more additional intervening materials such that contact is possible but not required. Neither “on” nor “over” implies any directionality as used herein.

The term “about” indicates that the value listed may be somewhat altered, as long as the alteration does not result in nonconformance of the process or structure to the illustrated embodiment. Finally, “exemplary” indicates the description is used as an example, rather than implying that it is an ideal. Other embodiments of the present teachings will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the present teachings being indicated by the following claims.

Terms of relative position as used in this application are defined based on a plane parallel to the conventional plane or working surface of a workpiece, regardless of the orientation of the workpiece. The term “horizontal” or “lateral” as used in this application is defined as a plane parallel to the conventional plane or working surface of a workpiece, regardless of the orientation of the workpiece. The term “vertical” refers to a direction perpendicular to the horizontal. Terms such as “on,” “side,” “higher,” “lower,” “over,” “top,” and “under” are defined with respect to the conventional plane or working surface being on the top surface of the workpiece, regardless of the orientation of the workpiece.